Planetary and Space Science 50 (2002) 123–128
www.elsevier.com/locate/planspasci
Collisional dissociation cross sections for O + O2, CO and N2,
O2 + O2, N + N2, and N2 + N2
R.E. Johnson ∗ , M. Liu, C. Tully
Engineering Physics Program and Astronomy Department, University of Virginia, Charlottesville, VA 22904-4745, USA
Received 7 February 2001; accepted 21 May 2001
Abstract
Using laboratory data for the scattering of O by O2 , CO and N2 pair potentials are constructed for O, C and N atoms. These potentials
are then used to calculate collisional dissociation and energy transfer cross sections for O on O2 , CO and N2 ; O2 on O2 , N on N2 , and N2
on N2 . Since the energetic ions often neutralize in the atmosphere corona, such cross sections are relevant to plasma ion bombardment
c 2002 Elsevier Science Ltd. All rights reserved.
of the atmospheres of Mars, Titan, Triton and Europa. 1. Introduction
The 4ow of the solar wind plasma, a plasma trapped in a
planetary magnetic 5eld or a local pick-up ion plasma onto
the exobase of an atmosphere produces chemistry, heating
and atmospheric loss. These processes, which can a9ect the
evolution of an atmosphere, are often referred to as atmospheric sputtering. When the atmosphere near the exobase
is atomic, both Monte Carlo descriptions (e.g., Johnson
et al., 2000) and analytic models (e.g., Johnson, 1990, 1994)
of the collision cascades initiated by the incident ions are
available. When measured atomic collision cross sections
are not available for such calculations, scaled potential functions have been used to describe collisional ejection.
Atmospheres on a number of small planetary bodies have
been shown to have molecules at the exobase and in the
corona: SO2 at Io (Wong and Johnson, 1996), CO2 at Callisto (Carlson, 1999) and Mars (Bougher et al., 1999), O2 at
Europa and Ganymede (Hall et al., 1995, 1998), N2 at Titan
(Yelle et al., 1997) and Triton (Cruikshank et al., 1993). In
such cases, atmospheric sputtering calculations using a fully
dissociated atmosphere can be used to roughly estimate the
loss (Johnson et al., 2000; Johnson and Luhmann, 1998). In
addition, if whole molecules are the principal ejecta, simple collision cross sections can be used (Pospieszalska and
Johnson, 1996). Monte Carlo atmospheric models which
correctly treat collisions that allow molecular dissociation
are not only more complicated, but in the energy range of
∗
Corresponding author. Tel.: +1-804-982-2335;
fax: +1-804-924-1353.
E-mail address: [email protected] (R.E. Johnson).
interest (∼a few keV to ∼ 20 eV) the required cross sections
are typically not available. Since the incident ions often
neutralize well above the exobase, the energetic particles are
often neutrals. There are essentially no laboratory data for
collisional dissociation of relevant molecules by energetic
neutrals.
In response to a Monte Carlo description of sputtering
of Mars’ atmosphere (Kass and Yung, 1995), we calculated collisional dissociation cross sections for energetic O,
CO and CO2 on CO2 using classical molecular dynamics
and scaled pair potentials (Johnson and Liu, 1998). In the
work described here, we extract semi-empirical potentials
from laboratory scattering data and use them to calculate
collisional energy transfer by energetic O to O2 ; N2 and
CO. That is, we use laboratory data on the scattering of energetic O atoms by molecules to construct pair potentials
between the incident and target atoms. Then we use these
potentials to calculate the energy transfer and collisional
dissociation cross sections. We also scale these results to
estimate the N + N2 and N2 + N2 cross sections needed at
Titan. In this paper, we do not treat dissociation due to electronic processes. These begin to become important at higher
energies and will be dealt with in the subsequent work. The
calculated cross sections are compared to those calculated
using the binary encounter model and their relevance for
Europa and Titan are discussed.
2. Empirical potentials
Smith et al. (1996), Lindsay et al. (1998) and Schafer
et al. (1987) have measured the angular di9erential scatter-
c 2002 Elsevier Science Ltd. All rights reserved.
0032-0633/02/$ - see front matter PII: S 0 0 3 2 - 0 6 3 3 ( 0 1 ) 0 0 0 6 7 - 8
124
R.E. Johnson et al. / Planetary and Space Science 50 (2002) 123–128
ing cross sections, (), for the atomic O on a number
of molecules (O2 ; N2 ; CO and CO2 ) in the energy range
0.5 –5 keV. These are presented as vs. plots, where
= sin () and = EO with EO and as the energy
and scattering angle of the incident O. In this form, the data
is nearly energy independent (e.g., Johnson, 1982). To obtain interaction potentials, we parameterize standard forms
for pair potentials between the incident atom and the atoms
in the molecules. These potentials are then used in a classical molecular dynamics program to calculate the scattering
cross sections. The parameters for each potential form are
adjusted to reproduce the – data. By creating interaction
potentials which reproduce the scattering data we are able
to accurately determine the total energy transfer to the target
molecule. Since the laboratory data do not give dissociation,
we use the constructed potentials to calculate the energy
transfer as well as to calculate dissociation cross sections.
In these calculations, a Morse potential
V (r) = De {exp[ − 2(r − r0 )] − 2 exp[ − (r − r0 )]};
(2.1)
is used to describe the interaction between the atoms
N
in the target molecule. Here, De = 5:19 eV; = 2:958= A
N
N
and r0 = 1:2 A for O2 and De = 9:8 eV; = 2:718= A and
N for N2 . For the N2 target, both accurate ground
r0 = 1:1 A
state N2 potential energy and the Morse potential were
used and a comparison of the calculated dissociation cross
sections was made. Although the Morse potential is not accurate near the dissociation limit, very little change (a few
percent) occurred in the collisional dissociation cross section and in the e9ective threshold for dissociation when the
accurate N2 ground state potential was used. It was found
that the dissociation cross section was primarily sensitive
to the size of the dissociation energy and to the interaction
potentials between the incident and target particles.
To describe the interaction of an O atom with the individual atoms of a diatomic molecule, a potential scaled to data
can be used. For collisions between atoms with open shells
there are a number of interaction potentials associated with
the atomic ground states. Although these a9ect the trajectories in low energy collisions (Tully and Johnson, 2001), at
the separations of primary interest here the interactions are
repulsive. The so-called universal repulsive potential
(Zeigler et al., 1985; hereafter the ZBL potential), which is
obtained from a 5t to scaled cross section data has been used
to describe the interaction potentials for the atomic systems
in this study. Here, we test it against the scattering data,
using it as the pair potential between the incident atom and
each atom in the molecule. The ZBL potential is given by
VZBL (r) =
Z A ZB e 2
’(r=aZBL );
r
aZBL = 0:8853a0 =(ZA0:23 + ZB0:23 );
(2.2)
Fig. 1. – plot for scattering of O from O2 . Laboratory data: solid triangles (Schafer et al., 1987; Smith et al., 1996, at a number of energies,
0.5 –5 keV) and Fuls et al. (1957) for energetic Ne+ + Ne, solid inverted
triangles. Calculations at two energies, 0.5 and 5 keV (results are nearly
identical until the maximum energy transfer is reached): open circles,
theory (Abrahamson, 1969); open triangles, ZBL potential (Zeigler
et al., 1985) ; open diamonds our favored potential [Vx−1 + Vn−1 ]−1
where n = 2 (Table 1). ( = sin (), where is the laboratory scattering angle and () is the laboratory scattering cross section and = E0 ,
where E0 is the incident atom energy.)
where
’(x) = 0:181 exp(−3:2x) + 0:5099 exp(−0:9423x)
+0:2802 exp(−0:4029) + 0:02817 exp(−0:2016x);
(2.3)
a0 is the Bohr radius and r is the internuclear separation.
More recently, the individual potentials of Gartner and Hehl
(1979) have been improved upon and used (J.F. Ziegler,
private communication) having a form for ’ like that of the
above, but with di9erent coePcients.
A simple exponential potential of the form
Vx (r) = Ax exp(−r=ax );
(2.4)
often called a Born–Mayer potential, is also examined.
Such forms have been used for atom–atom collisions and
as a pair potential for atom–molecule collisions (Macheret
and Adamovich, 2000). It is 5rst incorporated using parameters from Abrahamson (1969): Ax = 1316:1, 1709.9 and
2143:4 eV and (ax )−1 = 2:016, 2.009 and 2.005 (a0 )−1 ,
respectively, for C + C, N + N and O + O. For the heteronuclear collisions, the form V12 ≈ (V11 V22 )1=2 has been
tested and is recommended. Vx has also been extracted
from low energy data (Foreman et al., 1976). They give
N −1 for O+O and
Ax = 905; 316 eV and ax = 3:856; 3:221 A
O+N. Finally, the Born–Mayer form is also used here with
adjustable Ax and ax to describe the data at small (large r).
In Fig. 1 is given vs. calculated for O + O2 using
the ZBL potential and Vx from Abrahamson (1969) as pair
potentials between the incident O and an O in the molecule.
These are compared with the laboratory scattering data. It
is observed that VZBL is not a very good description of the
R.E. Johnson et al. / Planetary and Space Science 50 (2002) 123–128
interaction at any value of . Poor agreement with the data is
also found for all the other collision pairs. Similar results are
obtained using the individual potential for O + O, although
the agreement at small is better. Similarly, reducing the
screening constant for the VZBL (e.g., aZBL approaching 0.65
aZBL ) improved the agreement for O + O2 at small . The
convenient ZBL form for the potential has been used in
a number of calculations of atmospheric sputtering (e.g.,
Johnson and Liu, 1998). These are now being repeated with
better potentials. The potential Vx calculated for atom–atom
interactions is seen to give very good agreement at small
(large r) when used as a pair potential for the O + O2
collision, but fails dramatically at large .
To obtain more accurate interaction potentials we use analytical forms for the atom–atom pair potentials and 5t the experimental data for the collisions of interest. Recently, Tully
and Johnson (2001) calculated in detail low energy scattering of O with O. No separate experimental data exists for the
atom–atom interactions in the energy range of interest. For
scattering of O by O2 , CO and N2 , at large the dominant
scattering mode is a binary collision of the incident atom
with one of the atoms in the target, allowing the extraction
of an e9ective atom–atom potential, V (r). In this case the
measured is twice the size of for the atom–atom interactions. Here, we roughly extend the range of the data at large
for O + O by using the Ne+ +Ne data of Fuls et al. (1957).
If the atom–atom interaction potential is approximated by
a repulsive power law
Cn
Vn (r) = n
(2.5)
r
then the reduced cross section variable is given by
(Johnson, 1982)
2=n
1 an Cn
≈
;
(2.6)
n
where
an = ()(1=2)
[(n + 1)=2]
;
(n=2)
(2.7)
and (x) is the gamma function. Therefore, values of n and
Cn are easily extracted for a range of r (i.e., ) (e.g., Fuls
et al., 1957). At large , using half of the measured for the
collisions O+O2 the power law coePcients Cn and n are determined for each value of . We 5nd that a single value of n
and Cn can represent the data above ∼ 2 keV deg. At large
internuclear separations for O + O, the potential is nearly
exponential (implying n → ∞) and we use the form Vx
N
above. The range of data available is well 5t for ∼ 0:1–1 A
N
using Vn and at r & 1:2 A by 5tting Ax and ax for Vx ,
N and combine these in the
Ax = 2347:73 eV and ax = 0:2 A,
−1
−1 −1
form V (r) = [(Vn ) + (Vx ) ] as seen in Fig. 1.
These forms were also used to 5t the data for O + N2 and
O + CO. The calculated – 5ts to the laboratory data are
equivalent to those seen in Fig. 1 for O + O2 , and the appropriate parameters for the pair potentials are summarized
in Table 1. In order to construct the potentials for energetic
125
Table 1
Parameters for interaction potentials
Cn (eV · An )[Ax (eV)]
Potential
O + O2
(1). [ V1 + V1 ]−1 Vn = Crn
n
u
(2). Vx = Ax exp(−r=ax )
1
1 −1
(3). [ V + V ]
n
x
(4). Vn spline to Vx
O + N2
[ V1 + V1 ]−1
n
u
O + CO
1
1 −1
[V + V ]
n
u
O+O
(4)
Vn spline to Vx
O 2 + O2
[ V1 + V1 ]−1
n
u
N + N2
(5). Vn = Cn =r n
8.38
[2143.4]
[2347.7]
[905.0]
n[ax (A)]
2
[0.264]
[0.2]
[0:259]
4.48
2.59
4.44
2.68
8.38
2
2.465
3.301
(1) Vu : Universal (ZBL), Ziegler, J.F. et al. (1985). The stopping and
range of ions in solids. Pergamon, New York.
(2) Abrahamson, A.A. (1969) Born–Mayer-type interatomic potential
for neutral ground-state atoms with Z = 2 to 105, 178, 76 –79.
(3) Our best 5t of Ax and ax .
(4) Foreman, B.P. et al. (1976) used for Vx Repulsive potentials for
the interaction of oxygen atoms with the noble gases and atmospheric
molecules 12, 213–224.
(5) Also used [ V1 + V1 ]−1 using Cn = 2:4 eV An , n = 3:18 obtained
n
u
from above data using V12 ≈ [V11 × V22 ]1=2 and the new individual potential for Vu (Ziegler, J.F., personal communication). Resulting potential
is close to power potential given.
N incident on N2 we use the best power law potential extracted for O + O and O + N and the procedure described
in Abrahamson (1969) for pair potentials to construct an
N + N potential. This is used in calculations for N + N2 and
N2 + N2 . We compare two cases. First, we use the power
laws for O + O and O + N in Table 1 for large and combine with Vu for N + N using the parameters for the individual potential (Gartner and Hehl, 1979). We also use a
single power law 5t over the full range of for both O + O2
and O + N2 and obtain a simple power law potential for
N + N. The resulting potentials are very similar.
3. Cross sections
Using the above pair potentials in a classical molecular
dynamics calculation, the total energy transfer to the target molecule is calculated. The collisional dissociation cross
section is then calculated. Dissociation occurs when the internal energy of the diatom after the collision exceeds the
binding energy. For each initial molecular orientation and
for each impact parameter this is examined after the collision. If the total energy between the atoms is negative the
molecule remains intact, otherwise the diatom dissociates.
The probability of dissociation, pD , is obtained by counting
the number of times the molecule dissociates for a single
impact parameter and dividing by the number of molecular
126
R.E. Johnson et al. / Planetary and Space Science 50 (2002) 123–128
Fig. 2. Collisional dissociation cross sections vs. incident atom energy
obtained from pair potentials in Table 1. 1. O + N2 , 2. O + O2 , 3. N + N2 ,
4. O + CO, 5. O2 + O2 , and 6. N2 + N2 . Parameters of 5ts for these are
given in Table 2.
Table 2
Parameters for 5tting D [D = C(E − Et )x =(A + E y )]a
Parameters O + O2
O + CO O + N2
O 2 + O2
N2 + N2
N + N2
C
x
A
y
Et
4.69
1.34
9.96
1.58
29.0
8.53
1.42
2.06
1.68
14.5
3.01
1.11
0.749
1.20
19.9
3.38
1.73
1.16
1.53
24.9
4.51
1.03
0.21
1.31
14.5
2.75
0.96
2.25
1.18
29.0
a Energy is in eV; cross section is in 10−16 cm 2 ; E is an e9ective
t
‘threshold’ here: D → 0:05 − 0:1.
orientations. This process is repeated for a large number of
b to obtain the integrated cross sections, given by
pD bSb;
(3.1)
D = 2
where pD is the probability of dissociation averaged over
orientations.
Collisional dissociation cross sections are shown in
Fig. 2 for O + O2 , O + CO, O + N2 ; N + N2 ; O2 + O2
and N2 + N2 using the parameters for the pair potentials in
Table 1. As shown earlier for O + CO2 collisions (Johnson
and Liu, 1998), but not shown here, the oft-used binary
encounter approximation is reasonably accurate above the
steep rise in the cross section at low energy (the e9ective
threshold) but poorly describes the e9ective threshold. This
convenient model can also be used in the e9ective threshold region if a ‘dissociation energy’ (Sieveka and Johnson,
1984) 5tted to the e9ective threshold in Fig. 2 is used. It is
the transfer of energy to the target as a whole that results
in a larger than expected e9ective threshold even accounting for the mass ratios. For the atom–molecule collisions
calculated it is about three times the actual dissociation energy. Dissociation can occur down to the true threshold but
is not accurately calculated by the model used here. Also
shown are dissociation cross sections for collisions between
molecular species. For use in atmospheric models, in Table
2 we present parameters for 5ts to these cross sections.
Fig. 3. Di9usion (momentum transfer) cross section, d , calculated using
the pair potentials in Table 2. O + O2 (solid line), O2 + O2 (dash–dot),
N + N2 (dotted), and N2 + N2 (dash–dot–dot). O + O2 (dashed line)
using Sn =[Eo =2] as discussed in the text.
In Fig. 3 results are given for d , the di9usion (momentum
transfer) cross section, for O+O2 ; N+N2 , O2 +O2 and N2 +
N2 again using the potentials in Table 1. These are the cross
sections that control the escape of energetic particles from
a molecular planetary atmosphere (Johnson, 1994). Therefore, these determine the exobase altitude, ∼(nd )−1 . For
collisions in which energy is lost to internal degreesof freedom, the momentum transfer cross section is d =
(1 −
f cos )b db d, where is the center of mass scattering angle and is the azimuthal angle. This is then averaged over
molecular orientations as above. Here, f is the ratio of the
center of mass momentum after the collision to that before
the collision, which is unity when there is no energy lost
to internal degrees of freedom. Also shown for O + O2 is
Sn =(Eo =2) , where Sn is the energy transfer cross section and
Eo is the maximum energy transfer in a head-on collision
for point particles, = 4MA MB =(MA + MB )2 , where MA and
MB are the masses of the incident and target particles. This
is exactly equal to d for the collision of two atoms with no
internal degrees of freedom. For collision of an atom with
a diatom it is slightly larger than d . In analytical models
of the sputtering of atmospheres (Johnson, 1994), the ratio
Sn (Eo )=[Ud (U )]; where U is the gravitational binding energy and Eo is the incident ion energy, roughly determines
the number of molecules ejected per ion incident.
It is also seen that the dependence of d on Eo is somewhat di9erent for nitrogen than oxygen. Often the same
potential parameters are used for both O2 and N2 . Since the
nitrogen potentials were obtained by a less direct method
we will carry out a much more detailed set of calculations
for this system in the future using the full set of ground
state potentials, as we did earlier for O + O. The results for
R.E. Johnson et al. / Planetary and Space Science 50 (2002) 123–128
127
Fig. 5. The probability of a dissociation vs. impact parameter, b, for
1 keV (solid line) 200 eV (dashed), and 60 eV (dotted) O on O2 . Similar
results are obtained for N + N2 if one scales b via the dissociation cross
section in Fig. 2.
Fig. 4. (a) The average energy transfer, T , to the target molecule vs.
impact parameter, b, for O + O2 and (b) the fraction, Ecm =T , going into
internal motion (vibration and rotation) of the struck molecule. Results are
similar for N + N2 . 1 keV (sold line), 200 eV (dashed), 60 eV (dotted).
N2 + N2 and O2 + O2 are given at the energies relevant for
escape by energetic recoils from a molecular atmosphere.
Such cross sections have not been available to date and
have typically been assumed to be hard sphere cross sections having a constant size. This is seen to be incorrect.
Finally, for the escape of molecules from an atmosphere
dominated by atoms at the exobase, the cross sections for
atom–molecule collisions should be used, corrected by the
appropriate center of mass energy.
When modeling the e9ect of energetic ions or atoms incident on an atmosphere, either the energy transfer to the target
molecule vs. impact parameter or angular di9erential cross
sections are needed. Unfortunately, these forms are not independent of energy, but the results averaged over molecular
orientation do depend on energy in a simple way, as shown
in Fig. 4. In Fig. 4(a), the average energy transfer to the target molecule vs. impact parameter for O + O2 is displayed.
At a low energy, it has a dependence on b like that seen in
atom–atom collisions, but at 1 keV when binary encounters
dominate, a peak at the average apparent separation of the
atoms in the molecule is seen. In Fig. 4(b), the average fraction of the energy transfer to the target molecule going into
internal degrees of freedom is shown. This is only slowly
varying with b to some value of b where it decays rapidly.
These ratios are very similar for N + N2 collisions at the
same collision velocity. Finally, in Fig. 5 the probability of
dissociation vs. impact parameter is given, a quantity also
useful in modeling atmospheric sputtering. Over the energy
region for which the cross section in Fig. 2 is 4at or slowly
varying with incident energy, the probability of dissociation
vs. b also changes signi5cantly. At higher energies, collisional dissociation decreases and electronically-induced dissociation eventually dominates.
4. Summary
We have calculated collisional dissociation cross sections
for energetic O atoms incident on a number of molecules
that occur at the exobases of planetary and satellite atmospheres. In these calculations we use the – data (de4ection
of energetic incident O) to extract interaction potentials in
the form of pair potentials. These potentials are then used to
calculate the collisional dissociation cross sections. The extracted potentials di9er signi5cantly from the model potentials (Abrahamson 1969; Macheret and Adamovich, 2000),
from potentials extracted from low energy collision data,
from the so-called ‘universal’ (ZBL) pair potentials and
from the ‘individual’ potentials. When applying pair potentials, the useful binary encounter approximation (Sieveka
and Johnson, 1984; Johnson, 1990) is found to be valid well
above the e9ective threshold but overestimates the dissociation at low collision energies. This simple method can give
useful results if an arti5cial dissociation energy is used as
a 5tting parameter. This energy di9ers from the molecular
dissociation energy because it accounts for the energy transfer to the center of mass of the molecule at large separations
in slow collisions. We also found that the ZBL potentials,
which we had been using in the earlier work, may be valid
for atom–atom collisions but considerably overestimated the
energy transfer cross sections if used in atom–molecule collisions. Finally, at higher energies than those shown here,
electronically-induced dissociation must be included. Such
calculations are in progress.
128
R.E. Johnson et al. / Planetary and Space Science 50 (2002) 123–128
Due to the interest in Titan which has N2 at the exobase,
we used a rough model for obtaining the e9ective N + N
pair potential from the O + O and O + N pair potentials.
These are then used to calculate the N + N2 cross sections
given here. Cross sections for this system based on 5rst
principle are also in progress. Useful 5ts to the dissociation
cross sections are given in Table 2. Also given are the cross
section controlling energy deposition in an atmosphere (Sn )
and escape from the atmosphere (d ). These should be used
to compliment our recent accurate calculation of the escape
cross section for ground state collisions of O + O (Tully
and Johnson, 2001).
The results obtained here have recently been used in describing the collisional sputtering of the atmospheres of Titan (Shematovich et al., 2001) and Europa (Shematovich
and Johnson, 2001). At Europa, it was shown that Saur et
al. (1998) overestimated the energy transfer to atmospheric
O2 . Further, at Mars we used such calculations to correct
a model (Kass and Yung, 1995) which suggested that the
CO2 molecules at the exobase signi5cantly enhanced atmospheric sputtering yields (Leblanc and Johnson, 2001). The
data presented here is now available for additional detailed
modeling of atmospheric sputtering.
Acknowledgements
This work was supported by the NASA’s Planetary
Atmospheres Program.
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